Finite Group Databasebeta

← #10594359

Subgroups — #10594359

Order 576.

623 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3795C211yesmaximal, characteristic
2#64096C211yesmaximal, characteristic, complemented
3C3×#893C211yesmaximal, characteristic, complemented
4C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
5C4×#625C212yesmaximal, complemented
6#63898C211yesmaximal, characteristic, complemented
7#63879C212yesmaximal, complemented
8#63878C212yesmaximal, complemented
9D3×Dic3⋊C4C212yesmaximal, complemented
10#63855C211yesmaximal, characteristic, complemented
11#63844C211yesmaximal, characteristic, complemented
12#5546—31nomaximal
13#5504—31nomaximal
14C2×#634C2211yescharacteristic
15C2×C3×D12C2211yescharacteristic
16C2×C3×C4×D3C2212yes—
17C2×C3×C4×D3C2212yes—
18C2×C3×C4×D3C2211yescharacteristic
19C2×#625C2212yes—
20C2×D3×Dic3C2212yes—
21C2×D3×Dic3C2212yes—
22#3795C418yescomplemented
23C32×C4⋊C4C2211yescharacteristic
24#3750C2212yes—
25C4×C3⋊Dic3C2211yescharacteristic
26C4×C3⋊Dic3C2212yes—
27C3×(C2×D3)⋊C4C2212yes—
28C3×C4⋊Dic3C2211yescharacteristic
29C3×Dic3⋊C4C2212yes—
30C3×C4×Dic3C2211yescharacteristic
31#3724C2212yes—
32Dic3⋊Dic3C2211yescharacteristic
33Dic3⋊Dic3C2211yescharacteristic
34(C2×D3)⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic3C2211yescharacteristic
36(C2×D3)⋊Dic3C2212yes—
37Dic32C2212yes—
38C2×#237—31no—
39C2×#235—31no—
40C3×C4×D4—31no—
41C2×C3×C4⋊C4—31no—
42D4×Dic3—31no—
43C4×#67—32no—
44C4×#67—32no—
45C4×#67—32no—
46#926—31no—
47C2×Dic3⋊C4—32no—
48#893—31no—
49#893D311yescharacteristic, complemented
50#892—32no—
51D3×C4⋊C4—32no—
52D3×C4⋊C4D311yescharacteristic, complemented
53#887—31no—
54#887—31no—
55#887—31no—
56#887—61no—
57#67⋊C4—32no—
58#877—32no—
59#872—31no—
60C4×Dic6—31no—
61C22×C3×D3C2311yescharacteristic
62C22×C3×D3C2312yes—
63C2×C32×C4C2312yes—
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2312yes—
68#634C2×C414yescomplemented
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×D12C2×C414yescomplemented
74C3×C4×D3—42no—
75C3×C4×D3—42no—
76C3×C4×D3C2×C414yescomplemented
77#625C2×C418yescomplemented
78D3×Dic3—42no—
79D3×Dic3C2×C418yescomplemented
80#520—91noSylow
81C2×C3×D4—31no—
82C22×C3×C4—31no—
83C22×C3×C4—32no—
84C22×C3×C4—32no—
85C2×#67—32no—
86C2×#67—32no—
87C22×Dic3—32no—
88C22×Dic3—32no—
89#237—38no—
90#235—38no—
91C2×D12C2×D311yescharacteristic
92C2×C4×D3—32no—
93C2×C4×D3—32no—
94C2×C4×D3C2×D312yes—
95C2×C4×D3C2×D312yes—
96C2×C4×D3C2×D311yescharacteristic
97C2×Dic6—31no—
98C2×Dic6—31no—
99C2×Dic6—61no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—31no—
102C3×C4⋊C4—32no—
103C3×C4⋊C4C2×D311yescharacteristic
104C3×C4⋊C4C2×D311yescharacteristic
105C3×C22⋊C4—32no—
106C3×C42—31no—
107#217—32no—
108#217—31no—
109#217—31no—
110(C2×D3)⋊C4—31no—
111(C2×D3)⋊C4—31no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4C2×D312yes—
114C4⋊Dic3—31no—
115C4⋊Dic3—31no—
116C4⋊Dic3C2×D311yescharacteristic
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—32no—
120Dic3⋊C4—31no—
121Dic3⋊C4—31no—
122Dic3⋊C4—32no—
123Dic3⋊C4—62no—
124Dic3⋊C4C2×D312yes—
125C4×Dic3—31no—
126C4×Dic3—32no—
127C4×Dic3—32no—
128C4×Dic3—31no—
129C4×Dic3—32no—
130C4×Dic3—32no—
131C4×Dic3—62no—
132C4×Dic3—61no—
133C4×Dic3C2×D311yescharacteristic
134C22×C32C2411yescharacteristic
135C2×C3×D3—22no—
136C2×C3×D3—21no—
137C2×C3×D3—21no—
138C2×C3×D3—22no—
139C2×C3×D3C22×C414yes—
140C2×C3×D3C22×C412yes—
141C32×C4—22no—
142C32×C4C22×C412yes—
143C3⋊Dic3—21no—
144C3⋊Dic3—21no—
145C3⋊Dic3C22×C414yes—
146C3×Dic3—22no—
147C3×Dic3—21no—
148C3×Dic3—21no—
149C3×Dic3C22×C412yes—
150C2×#41—91no—
151C4×Q8—91no—
152C4×Q8—91no—
153C4×D4—31no—
154C4×D4—92no—
155C4×D4—91no—
156C4×D4—92no—
157#117—91no—
158#117—92no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—92no—
161C23×C3—31no—
162C23×C3—32no—
163C22×D3C22×D311yescharacteristic
164C22×D3C22×D312yes—
165C3×D4—34no—
166C2×C3×C4—21no—
167C2×C3×C4—22no—
168C2×C3×C4—31no—
169C2×C3×C4—34no—
170C2×C3×C4—31no—
171C2×C3×C4—32no—
172C2×C3×C4—31no—
173C2×C3×C4—122no—
174C2×C3×C4—122no—
175C2×C3×C4C22×D312yes—
176C2×C3×C4C22×D312yes—
177C2×C3×C4C22×D311yescharacteristic
178C2×C3×C4C22×D311yescharacteristic
179#67—38no—
180#67—38no—
181C2×Dic3—32no—
182C2×Dic3—32no—
183C2×Dic3—38no—
184C2×Dic3—31no—
185C2×Dic3—31no—
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—61no—
189C2×Dic3—62no—
190C2×Dic3—61no—
191C2×Dic3—122no—
192C2×Dic3C22×D311yescharacteristic
193C2×Dic3C22×D312yes—
194C2×Dic3C22×D311yescharacteristic
195C2×Dic3C22×D311yescharacteristic
196D12C4×D314yescomplemented
197C4×D3—38no—
198C4×D3—42no—
199C4×D3—42no—
200C4×D3—122no—
201C4×D3C4×D314yescomplemented
202Dic6—34no—
203Dic6—34no—
204Dic6—64no—
205C2×C32C23×C411yescharacteristic
206C2×C32H(5,𝔽2)11yescharacteristic
207C2×C322−1+411yescharacteristic
208C3×D3—22no—
209C3×D3—24no—
210#41—98no—
211C2×Q8—91no—
212C2×D4—31no—
213C2×D4—92no—
214C22×C4—31no—
215C22×C4—32no—
216C22×C4—32no—
217C22×C4—92no—
218C22×C4—92no—
219C4⋊C4—32no—
220C4⋊C4—31no—
221C4⋊C4—91no—
222C4⋊C4—92no—
223C4⋊C4—91no—
224C4⋊C4—92no—
225C4⋊C4D3211yescharacteristic
226C22⋊C4—32no—
227C22⋊C4—91no—
228C22⋊C4—91no—
229C22⋊C4—92no—
230C42—31no—
231C42—92no—
232C42—92no—
233C42—91no—
234C22×C3—21no—
235C22×C3—34no—
236C22×C3—32no—
237C22×C3—62no—
238C22×C3—61no—
239C22×C3—62no—
240C22×C3—61no—
241C22×C3C23×D311yescharacteristic
242C22×C3C23×D311yescharacteristic
243C2×D3—21no—
244C2×D3—22no—
245C2×D3—21no—
246C2×D3—22no—
247C2×D3C2×C4×D314yes—
248C2×D3C2×C4×D312yes—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—32no—
253C3×C4—42no—
254C3×C4—62no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4C2×C4×D312yes—
258C3×C4C2×C4×D312yes—
259Dic3—21no—
260Dic3—22no—
261Dic3—21no—
262Dic3—34no—
263Dic3—34no—
264Dic3—64no—
265Dic3—62no—
266Dic3—61no—
267Dic3—61no—
268Dic3—61no—
269Dic3—61no—
270Dic3—121no—
271Dic3C2×C4×D312yes—
272C32#52011yescharacteristic, complemented, Sylow
273C23—32no—
274C23—31no—
275Q8—94no—
276D4—34no—
277D4—98no—
278C2×C4—31no—
279C2×C4—34no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—32no—
283C2×C4—98no—
284C2×C4—92no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—362no—
290C2×C4C2×D3212yes—
291C2×C4C2×D3211yescharacteristic
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—64no—
296C2×C3—62no—
297C2×C3C22×C4×D311yescharacteristic
298C2×C3C22×C4×D311yescharacteristic
299C2×C3#100411yescharacteristic
300C2×C3#100411yescharacteristic
301C2×C3#100711yescharacteristic
302C2×C3#101011yescharacteristic
303D3—22no—
304D3—24no—
305C22—32no—
306C22—34no—
307C22—61no—
308C22—62no—
309C22—62no—
310C22—61no—
311C22C22×D3211yescharacteristic
312C4—22no—
313C4—32no—
314C4—61no—
315C4—62no—
316C4—61no—
317C4—94no—
318C4—181no—
319C4—181no—
320C4C4×D3212yes—
321C3—21no—
322C3#550411yescharacteristic, complemented
323C3#554611yescharacteristic, complemented
324C2—62no—
325C2—64no—
326C2C2×C4×D3211yescharacteristic
327C2#6431311yescharacteristic
328C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutatorC2×C32C23×C4
FittingC32×C4⋊C4C22
socle, cosocleC22×C32C24
prime radicalC2×#3795C2
hypercenterC4⋊C4D32
hypocenterC32#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #64096C2semidirect
3. C3×#893C2semidirect
4. C3×D3×C4⋊C4C2semidirect
5. C4×#625C2semidirect
6. #63898C2semidirect
7. #63879C2semidirect
8. #63878C2semidirect
9. D3×Dic3⋊C4C2semidirect
10. #63855C2semidirect
11. #63844C2semidirect
22. #3795C4semidirect
49. #893D3semidirect
52. D3×C4⋊C4D3semidirect
68. #634C2×C4semidirect
73. C3×D12C2×C4semidirect
76. C3×C4×D3C2×C4semidirect
77. #625C2×C4semidirect
79. D3×Dic3C2×C4semidirect
196. D12C4×D3semidirect
201. C4×D3C4×D3semidirect
272. C32#520semidirect
322. C3#5504semidirect
323. C3#5546semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#37952
#640962
C3×#8932
C3×D3×C4⋊C42
C4×#6252
#638982
#638792
#638782
D3×Dic3⋊C42
#638552
#638442
#55463
#55043